J. of Math. (PRC) 6 n (nt ) + n V = 0, (1.1) n t + div. div(n T ) = n τ (T L(x) T ), (1.2) n)xx (nt ) x + nv x = J 0, (1.4) n. 6 n

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1 Vol. 35 ( 215 ) No. 5 J. of Math. (PRC) a, b, a ( a. ; b., 4515) :., [3]. : ; ; MR(21) : 35Q4 : O175. : A : (215) [1] : [ ( ) ] ε 2 n n t + div 6 n (nt ) + n V =, (1.1) n div(n T ) = n τ (T L(x) T ), (1.2) λ 2 V = n C(x), (1.3) n, T V ; C(x) T L (x) ; ε >, τ > λ >. (1.1) (1.3) [1]. [1] (1.1) (1.3), [2]. [3] (1.1) (1.3),, : ( ) ( 6 n n)xx (nt ) x + nv x = J, (1.4) n x (nt x ) x = n τ (T L(x) T ), (1.5) λ 2 V xx = n C(x) in (, 1), (1.6) n() = n(1) = 1, n x () = n x (1) =, T () = T, T x () = T x (1) =, (1.7) V () = V = ε2 6 ( n) xx () + T, (1.8) : : (12A11); 213 ( ); 213 ( ); (213ZD556) (213GGJS-142). : (198 ),,,, :.

2 16 Vol. 35 J, : 1.1 ( [3]) C(x), T L (x) L (, 1), C(x) >, < m L T L (x) M L, x (, 1), (1.4) (1.8) (n, T, V ) n(x) e M >, x (, 1), M M = e 2M τm 2 L (M L m L )M L + 2(e 1 + C(x) log C(x) L (,1)) λ 2 m L (1.9). (1.4) (1.8), , m L J, (1.4) (1.8). 1.1 (1.1) (1.3),, [4 6]. 1.2 T L (x) T, (1.1) (1.3) -, [7 19] [3] n = e u (1.4) (1.8) ( ) u xx + u2 x T xx (u x T ) x + eu C(x) = J 12 2 λ xx 2 (e u ) x, (2.1) (e u T x ) x = eu τ (T L(x) T ), (2.2) ( ) V (x) = ε2 u xx + u2 x (x) + T (x) + u x (s)t (s)ds + J e u(s) ds, (2.3) 12 2 u() = u(1) =, u x () = u x (1) =, T () = T, T x () = T x (1) =, (2.4) V () = V = ε2 12 u xx() + T, (2.5) (u, T, V ) H 4 (, 1) H 2 (, 1) H 2 (, 1), 1.2, (2.1), (2.2), (2.4) , m L J, m L εm 2 6mL J e M e2m ( 2ε + 4 6m L M)( 2e 2M + 1)(M L m L ) 2 2 >, (2.6) 2ε τ (2.1), (2.2), (2.4) (u, T ) H 4 (, 1) H 2 (, 1). 2.1 (1.9), m L M, m L J (2.6).

3 No. 5 : 17 [3] u xx 2 L 2 (,1) +m L 2 u x 2 L 2 (,1) e2m (M L m L )M L + λ 2 (e 1 + C(x) log C(x) L (,1)), 2τm L (2.7) u L (,1) M, (2.8) < m L T M L, (2.9) (2.9) [3] , 2.1 (u, T ) H 4 (, 1) H 2 (, 1) (2.1), (2.2), (2.4), u x L (,1) 4 6mL M ε, (2.1) T x L (,1) e2m τ (M L m L ). (2.11) (2.7), ml 6 ε u xx L2 (,1) u x L2 (,1) ε2 12 u xx 2 L 2 (,1) +m L 2 u x 2 L 2 (,1) e2m 2τm L (M L m L )M L + λ 2 (e 1 + C(x) log C(x) L (,1)), Hölder (1.9), u 2 x(x) = 2 u xx (s)u x (s)ds 2 u xx L (,1) u 2 x L 2 (,1) 2 [ ] 6 e 2M (M L m L )M L + λ 2 (e 1 + C(x) log C(x) L (,1)) ml τm L = 2 6 ml ε M 2 ml 2 = 6mL M 2, ε (2.1). (2.2) (, x), T x = e u τ e u (T L (x) T )dx, (2.8) (2.9) (2.11). 2.1 (u 1, T 1 ), (u 2, T 2 ) H 4 (, 1) H 2 (, 1) (2.1), (2.2), (2.4). T 1 T 2 (e u1 T 1x ) x = eu1 τ (T L(x) T 1 ) (e u2 T 2x ) x = eu2 τ (T L(x) T 2 )

4 18 Vol. 35, + 1 τ + 1 τ (2.8) e u1 (T 1 T 2 ) 2 xdx = (T L (x) T 2 )(e u1 e u2 )(T 1 T 2 )dx T 2x (e u1 e u2 )(T 1 T 2 ) x dx T 2x (e u1 e u2 )(T 1 T 2 ) x dx 1 τ e u1 (T 1 T 2 ) 2 dx (T L (x) T 2 )(e u1 e u2 )(T 1 T 2 )dx. (2.12) (2.8) e u1 (T 1 T 2 ) 2 xdx e M (T 1 T 2 ) 2 xdx. (2.13) e u1 e u2 e M u 1 u 2, (2.11), Hölder Poincaré, T 2x (e u1 e u2 )(T 1 T 2 ) x dx e2m τ (M L m L ) e M u 1 u 2 (T 1 T 2 ) x dx [ ] 1 [ e3m 2 1 τ (M L m L ) (u 1 u 2 ) 2 dx (T 1 T 2 ) 2 xdx [ ] 1 [ e3m 2 1 (M L m L ) (u 1 u 2 ) 2 xdx (T 1 T 2 ) 2 xdx. (2.14) 2τ (2.9),, 1 τ (2.12) (2.15) [ (T 1 T 2 ) 2 xdx u 1 u 2 12 (T L (x) T 2 )(e u1 e u2 )(T 1 T 2 )dx em 2τ (M L m L ) ( u 1xx + u2 1x 2 [ ] 1 [ 2 1 (u 1 u 2 ) 2 xdx e2m ( 2e 2M + 1)(M L m L ) 2τ ) xx [ T 1xx (u 1x T 1 ) x + eu1 C(x) λ 2 (T 1 T 2 ) 2 xdx. (2.15) (u 1 u 2 ) 2 xdx. (2.16) = J (e u1 ) x ( ) u 2xx + u2 2x C(x) T 2xx (u 2x T 2 ) x + eu2 = J 12 2 λ xx 2 (e u2 ) x

5 No. 5 : 19, (u 1 u 2 ) 2 xxdx + ε2 T 1 (u 1 u 2 ) 2 xdx + (u 2 1x u 2 2x)(u 1 u 2 ) xx dx + (T 1 T 2 ) x (u 1 u 2 ) x dx u 2x (T 1 T 2 )(u 1 u 2 ) x dx + 1 (e u1 e u2 )(u λ 2 1 u 2 )dx = J (e u1 e u2 )(u 1 u 2 ) x dx. (2.17) (2.1) Young, (u 2 1x u 2 2x)(u 1 u 2 ) xx dx = ε2 (u 1x + u 2x )(u 1 u 2 ) x (u 1 u 2 ) xx dx ε m L M (u 1 u 2 ) x (u 1 u 2 ) xx dx 12 ε2 (u 1 u 2 ) 2 xxdx εm 2 1 6mL (u 1 u 2 ) 2 xdx. (2.18) Hölder (2.16), [ ] 1 [ 2 1 (T 1 T 2 ) x (u 1 u 2 ) x dx (T 1 T 2 ) 2 xdx (u 1 u 2 ) 2 xdx (2.9) e2m ( 2e 2M + 1)(M L m L ) 2τ (u 1 u 2 ) 2 xdx. (2.19) T 1 (u 1 u 2 ) 2 xdx m L (u 1 u 2 ) 2 xdx. (2.2) (2.1), Hölder, Poincaré (2.16), 4 6mL M 1 u 2x (T 1 T 2 )(u 1 u 2 ) x dx T 1 T 2 (u 1 u 2 ) x dx ε 4 [ 6mL M 1 ] 1 [ 2 1 (T 1 T 2 ) 2 xdx (u 1 u 2 ) 2 xdx 2ε 4 6mL M e 2M ( 2e 2M + 1)(M L m L ) 2 2ε τ e x 1 λ 2, Hölder Poincaré, (u 1 u 2 ) 2 xdx. (2.21) (e u1 e u2 )(u 1 u 2 )dx. (2.22) J (e u1 e u2 )(u 1 u 2 ) x dx J e M u 1 u 2 (u 1 u 2 ) x dx J e M 1 (u 1 u 2 ) 2 xdx. (2.23) 2

6 125 Vol. 35 (2.17) (2.23) (u 1 u 2 ) 2 xxdx + C (u 1 u 2 ) 2 xdx, (2.) C = m L εm 2 6mL J e M e2m ( 2ε + 4 6m L M)( 2e 2M + 1)(M L m L ) ε τ (2.) (2.6) u 1 = u 2, (2.16) T 1 = T 2. [1] Jüngel A, Mili sić J P. A simplified quantum energy-transport model for semiconductors [J]. Nonlinear Analysis: Real World Applications, 211, 12: [2] Chen L, Chen X Q, Jüngel A. Semiclassical limit in a simplified quantum energy-transport model for semiconductors [J]. Kinetic and Related Models, 211, 4: [3] Dong J W, Zhang Y L, Cheng S H. Existence of classical solutions to a stationary simplified quantum energy-transport model in 1-dimensional space [J]. Chin. Ann. Math., 213, 34(5): [4] Grubin H, Kreskovsky J. Quantum moment balance equations and resonant tunnelling structures [J]. Solid-State Electr., 1989, 32: 171. [5] Degond P, Gallego S and Méhats F. On quantum hydrodynamic and quantum energy transport models [J]. Commun. Math. Sci., 27, 5: [6] Chen R C and Liu J L. A quantum corrected energy-transport model for nanoscale semiconductor devices [J]. J. Comput. Phys., 25, : [7] Ju Q C, Chen L. Semiclassical limit for bipolar quantum drift-diffusion model [J]. Acta Mathematica Scientia, 29, 29B(2): [8] Chen X Q, Chen L, Jian H Y. Existence, semiclassical limit and long-time behavior of weak solution to quantum drift-diffusion model [J]. Nonlinear Analysis: Real World Applications, 29, 1(3), 1(3): [9] Jüngel A, Violet I. The quasineutral limit in the quantum drift-diffusion equations [J]. Asymptotic Analysis, 27, 53(3): [1] Chen L, Ju Q C. Existence of weak solution and semiclassical limit for quantum drift-diffusion model [J]. Z. Angew. Math. Phys., 27, 58: [11] Chen X Q, Chen L, Jian H Y. The Dirichlet problem of the quantum drift-diffusion model [J]. Nonlinear Analysis, 28, 69: [12] Chen X Q, Chen L. The bipolar quantum drift-diffusion model [J]. Acta Mathematica Sinica, 29, 25(4): [13] Chen X Q. The global existence and semiclassical limit of weak solutions to multidimensional quantum drift-diffusion model [J]. Advanced Nonlinear Studies, 27, 7: [14] Chen X Q, Chen L. Initial time layer problem for quantum drift-diffusion model [J]. J. Math. Anal. Appl., 28, 343: [15] Chen X Q. The isentropic quantum drift-diffusion model in two or three space dimensions [J]. Z. angew. Math. Phys., 29, 6(3):

7 No. 5 : 1251 [16] Chen X Q, Chen L, Jian H Y. The existence and long-time behavior of weak solution to bipolar quantum drift-diffusion model [J]. Chin. Ann. Math., 27, 28B(6): [17] Chen L, Ju Q C. The semiclassical limit in the quantum drift-diffusion equations with isentropic pressure [J]. Chin. Ann. Math., 28, 29B(4): [18] Abdallah N B, Unterreiter A. On the stationary quantum drift-diffusion model [J]. Z. Angew. Math. Phys., 1998, 49: [19] Nishibata S, Shigeta N, Suzuki M. Asymptotic behaviors and classical limits of solutions to a quantum drift-diffusion model for semiconductors [J]. Math. Models Methods Appl. Sciences, 21, 2(6): UNIQUENESS OF STATIONARY SOLUTIONS TO ONE-DIMENSIONAL QUANTUM ENERGY-TRANSPORT MODEL FOR SEMICONDUCTORS DONG Jian-wei a, ZHANG You-lin b, CHENG Shao-hua a (a. Department of Mathematics and Physics; b. Library, Zhengzhou Institute of Aeronautical Industry Management, Zhengzhou 4515, China) Abstract: In this paper, we study the classical solutions to the stationary quantum energytransport model for semiconductors in one space dimension. The uniqueness of the solutions is proved when the lattice temperature is sufficiently large and the current density is relatively small by using some inequality techniques, which is not obtained in [3]. Keywords: quantum energy-transport model; stationary solutions; uniqueness 21 MR Subject Classification: 35Q4

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